Local rigidity of weak or no hyperbolicity algebraic actions

TL;DR

This paper establishes $C^ abla$ local rigidity for a broad class of parabolic algebraic actions on homogeneous spaces, extending the understanding of rigidity properties in weak or non-hyperbolic settings.

Contribution

It introduces a new general strategy for proving $C^ abla$ local rigidity of algebraic actions with weak or no hyperbolicity, addressing a previously unexplored area.

Findings

Proves $C^ abla$ local rigidity for parabolic algebraic actions

Develops a new strategy for rigidity proofs in non-hyperbolic contexts

First demonstration of strong local rigidity for these actions

Abstract

In this paper we study rigidity properties of abelian \hyphenation{break-able}actions with weak or no hyperbolicty. We introduce a general strategy for proving $C^\infty$ local rigidity of algebraic actions. As a consequence, we show $C^\infty$ local rigidity for a broad class of parabolic algebraic actions on homogeneous spaces of semisimple Lie groups. This is the first time in the literature that (strong) local rigidity for these actions is addressed.